SOLUTION: Raleigh, NC is about 120 miles inland from the nearest point on the Atlantic coast; Savannah, Ga is about 300 miles south of that point down the coast. Assuming the coast is a stra

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Question 162169: Raleigh, NC is about 120 miles inland from the nearest point on the Atlantic coast; Savannah, Ga is about 300 miles south of that point down the coast. Assuming the coast is a straight line going north-south, C is the point along the coast directly east of Raleigh. It costs 2 cents per mile to transport a ton of shrimp by truck and 1 cent per mile to transport them by ship. The Savannah Shrimp company wants to find the point, P, on the Atlantic Coast to which it should truck its shrimp before loading then aboard to cargo ship in order to minimize the total cost of transporting them from Savannah to Raleigh.
a. Set up the function which the Savannah Shrimp Company must minimize.
b. Find the position of P which minimizes cost.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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Raleigh, NC is about 120 miles inland from the nearest point on the Atlantic coast; Savannah, Ga is about 300 miles south of that point down the coast. Assuming the coast is a straight line going north-south, C is the point along the coast directly east of Raleigh. It costs 2 cents per mile to transport a ton of shrimp by truck and 1 cent per mile to transport them by ship. The Savannah Shrimp company wants to find the point, P, on the Atlantic Coast to which it should truck its shrimp before loading then aboard to cargo ship in order to minimize the total cost of transporting them from Savannah to Raleigh.
a. Set up the function which the Savannah Shrimp Company must minimize.
:
Let s = distance by ship (Savannah to P)
Draw a diagram of this, note the truck distance to P will be the the hypotenuse
of a right triangle formed by the legs 120 mi and (300-s)
:
Cost = truck cost + ship cost
C = .02sqrt%28120%5E2+%2B+%28300-s%29%5E2%29 + .01s
:
:
b. Find the position of P which minimizes cost.
Using a graph, x= ship dist; y = cost mi/ton

Min cost s = 231 mi by ship,
:
P = 231 mi from Savannah
:
Find the dist by truck
T = sqrt%28120%5E2+%2B+69%5E2%29
T = 138.42 mi by truck